1. Field of the Invention
The present invention relates to a torsional vibration monitoring method for a rotating shaft system of apparatuses such as a turbine generator, in which torsional vibrations produced in the rotating shaft system are measured at a small number of measurable points on the system and torsional vibrations at arbitrary points on the same are estimated by linearly decomposing the measured vibrations. The invention relates particularly to a torsional vibration monitoring method of the type mentioned, in which up to 2n degrees of the vibrations at the arbitrary points are obtained from the detected torsional vibrations, so as to achieve a high precision torsional vibration monitoring method.
2. Description of the Prior Art
It has been known that it is important for designers to know various disturbances introduced into a rotating shaft system of apparatuses such as a turbine generator, compressor and marine diesel engine and, particularly, that it is important for operators to know the fatigue life of the rotating shaft system.
The rotating shaft system for a turbine generator is, however, very long and may be as long as several tens of meters. Therefore, it generally requires several measuring points to measure the disturbances along the shaft system.
Furthermore, it is important to measure the torsional vibrations produced in the rotating shaft system because they may damage the rotating shaft due to fatigue. However, it is economically disadvatageous to provide a number of torsional vibration measuring devices along the shaft and sometimes it is impossible to do so physically.
According to vibration theory, the torsional vibration Y(x,t) produced in the rotating shaft system is represented by a sum of modal vibrations Y.sub.i (x,t) and each modal vibration is represented by a product of vibration mode type G.sub.i (x) and vibration mode component H.sub.i (t), as follows: EQU Y(x,t)=.SIGMA.Y.sub.i (x,t) (1) EQU .SIGMA.G.sub.i (x).multidot.H.sub.i (t) (2)
The vibration mode type G.sub.i (x) is as shown in FIG. 1 and is previously determined dependent on a specific shaft system, .parallel.G.sub.i (x).parallel..apprxeq.o when i .fwdarw..infin..
By selecting i as a suitable value n, equation (2) can be rewritten as follow: ##EQU1## Therefore, n sets of torsional vibrations (See FIG. 2A) measured at n positions pk (coordinate being x.sub.pk, k=1, . . . , n) on the rotating shaft system are represented as follow: ##EQU2##
As mentioned, since each of the vibration mode types G.sub.i (x.sub.pk) has a constant value, the equation (3) ' can be referred to as the n-th dimension simultaneous equation of the n-th degree each having the vibration mode component H.sub.i (t) as a variable. Therefore, it is possible to obtain the reverse matrix components G.sub.ik ' (i and k are column and row, respectively) of a matrix containing n x n components G.sub.i (x.sub.pk) (i and k are row and column, respectively).
Accordingly, the i-th vibration mode component can be represented as EQU H.sub.i (t)=.SIGMA.G.sub.ik '.multidot.Y(x.sub.pk,t) (4)
and thus the torsional vibration at an arbitrary position j on the shaft system (coordinate being x.sub.j) can be estimated as follow: ##EQU3##
However, this method of estimating the torsional vibration at the arbitrary position j has a disadvantage in that , when it is assumed that the torsional vibration of the rotating shaft system is an accumulation of the modal vibration up to the n-th degree, the vibration must be measured at the corresponding number, i.e., n, of points on the shaft.
Where the torsional vibration is picked-up by using a pick-up device having physical dimensions, it is generally desirable to reduce the number of the pick-up devices to be used. On the contrary, it is desired to increase the number of the pick-up devices in order to increase the precision of the estimation.